Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Why teach geometry (part 2)
Posted:
Jun 4, 1994 7:44 PM
|
|
The sad fact is, I was a star geometry student in high school, and honestly
remember *very* few of the theorems we worked on: probably fewer than 10%.
(It
sharpened my deductive logic skills, though!) In my later years as a
computer
scientist, I became interested in computational geometry and analytic
geometry,
ie, geometry with numbers included!
I'm a firm believer in a visual approach to mathematics, particularly in
graphing functions, etc. Geometry seems like a natural lead-in to
visualizing
math. For example, you can demonstrate and prove the pythagoream theorem
by
taking 4 similar right triangles and arranging them in a square with a hole
inthe middle. Or, you can show that a^2 - b^2 = (a-b)(a+b), or that
(a-b)^2=
a^2-2*a*b+b^2, etc. Using geometric shapes to prove these theorems (as I
suspect the Greeks et al. did) is just another rich avenue for learning.
In
particular, it grounds the abstract algebraic manipulations in something
manipulative and visual.
I wouldn't want to eliminate geometry entirely, but I might change the
mixture
to make it more relevant to both college-bound and non-college-bound
students.
For instance, take a city map that's a grid of N-S and E-W streets. You're
in
a car on the NW corner, and want to drive to some location on the East
side.
You can show that ANY path you take, as long as you drive either South or
East,
toward your destination is the same length! This is a little
counter-intuitive
to students who memorized "the shortest distance between two points is a
straight line". Car's can't drive in straight lines through a city,
though.
Sorry for the long-windedness. This is the first time I've tried to put
these
thoughts on paper. Overall, I'd say we should study geometry (and not just
the
compass-straight-edge variety!) because:
1. It's an intellectually honest, important component of the development
of
mathematics.
2. One can prove a variety of algebraic, trig, and other properties using
geometric constructions
3. It's one of many possible vehicles for demonstrating deductive logic.
4. Many "real world" problems can be modeled as geometric problems.
My criticisms of current geometry practices are:
1. It's not the *only* method for demonstrating proof. In fact, we might
mis-lead students into thinking that only in geometry to we prove theorems;
the
rest of mathematics was created by black magic.
2. The compass-straight-edge model is an interesting intellectual
exercise,
but is *not* the sole essence of geometry. In particular, we should be
combining algebra and geometry at an earlier stage.
As far as the "what use is it" questions, some of it's useful for modelling
problems, and otherwise it's an intellectual strengthener. What "use" is
an
aerobics class? I mean, how often in the real world are you going to swing
your arms and legs in that manner? Or bench-pressing? How often do you
really
have to lift a load of iron off your chest? See what I mean? I'm cautious
about taking a too literal view of "what use is this" when it comes to
mathematics. On the other hand, we do aerobics and lifting so that we can
enjoy the other activities in our life more fully. Same should be true
(intellectually) with mathematics.
Good luck with your discussion! I, too, would be interested in a
synthesized
summary of these responses.
Thank you for asking a provocative question,
Larry Gallagher
Institute for Research on Learning
larry_gallagher@irl.com
This whole geometry debate has centered on the content of a Geometry course
-- Euclidean vs. non-Euclidean vs 3-dimensional, etc. etc. I would like to
point out an advantage that geometry offers in terms of the PROCESS of
learning.
One pleasant effect of our geometry curriculum here is that it provides an
excellent environment to introduce our students to the principles of
inductive reasoning and cooperative learning and problem-solving. Because
much of geometry can be done intuitively, it gives students a chance to
focus
a bit more closely on the social skills of classroom interaction. By the
time
they get to Precalculus and Calculus (when, one could argue, they NEED to
be
able to work together and reason inductively), their cooperative learning
skills are honed to the point that they work together smoothly (and often
regardless of whether the teachers asks them to or not!).
See Michael Serra's "Discovering Geometry: an Inductive approach" from Key
Curriculum Press for more information.
Subject: Re: Why teach Geometry in HS?
In addition to geometry being useful in certain `real world' situations. I
think that the subject, if taught correctly, can be used to develop
creative and critical thinking skills. Analyzing and synthesizing
information,
and making conjectures about geometric principals will force students to
think -which is something they don't do enough of as they coast through
classes imitating and regurgitating (sp?). Geometry students should be
active learners, and
much of it can be done cooperatively as well.
The Geometer's Sketchpad is a great piece of software to help in these
aims.
John, UNM
On Wed, 30 Mar 1994, Doug Hale wrote:
> Regarding college level algebra/geometry -- hear! hear!
> Doug
Dear Doug et al,
I have many notions that I would like to discuss on this issue. This
point of why not a college level algebra/geometry is a rather touchy one.
We are most fortunate to have a real *geometer* at NMSU; only ONE! My
initial educational experience was in aeronautical engineering followed by
YEEKS *PHILOSOPHY*!! Please note that it was the philosophy that brought
me home to mathematics. Yes that's right! Maybe as math type people we
can see the connection, but the formal world of education has lost the
vision of what philosophy began as...
Logical rhetoric is more than linguistics, Greek, and political
interpretations in the marketplace. It is indeed the foundation of
mathematical thought. When I interviewed departments at the University
to conduct my Master's Degree, the department of Education laughed at my
background. Their response "hmmm... Philosophy is not a teaching field!"
Whatever happened to the historical background of education. Did they
forget that PhD translates to Doctorate of Philosophy?? Maybe, they just
were never told what they were earning. Anyway... The department of
Mathematics did not laugh. They said "Wow, welcome aboard. Have you
studied the tenets of formal logic?" I said, "Of course, Why???" They
said,
"Well Kathrine, you should be able to master Boolean Algebra rather well!"
I said, "What's that? But I should tell you that I really understand
Geometry which I have studied as a logical set that defines the
characteristics of space." They said, "See Dr. Zund, He's our geometer!"
So, I did. And was I ever glad. You have not taken geometry until you
have taken a course from a real geometer. His specialty by the way, is in
topology. But most of all, he was a teacher! We even went through high
school texts and found geometric falacies in the proofs. I entered the
world of secondary math education as a total sceptic. That scepticism has
been reinforce over and over. We talk about the math anxiety of Elementary
Educators, but finding someone who really wants to teach high school
geometry is like pulling teeth.
Now, for the rest of the story...
I believe...
*We must love geometry and appreciate its logical and physical form in
order to teach it to kids. Otherwise we will continue to reinforce its
difficulty rather than its elegance of form, and thelogical thinking that
it promotes.
*We must recognize that the roots of Algebra and its *raison d'etre* is to
analyze geometric relationships. (Of course this is also true of Calculus
which is merely the extension of Algebraic Analysis of more advanced
space.)
*We must introduce our students to the difference in mathematical
disciplines, which I delienate as either Numerical or Geometric. Often as
mathematics people we mistakenly discuss this issue as the debate between
Pure and Applied Mathematics. This is a false discussion, because there
are both Pure and Applied mathematics in both the Numerical and Geometric
disciplines.
*In accordance with the above, we must recognize the diversity of the
Geometric discipline and its many spinoffs; e.g. fractals, manifolds,
non-Euclidian space, topology (and its little sister, Knots) ...
I could go on and on! Next year, I will be using the Creative
Publications Algebra Themes Tools and Concepts Curriculum. It proposes
that the learning of Algebra should take place in a lab environment where
manipulatives model the equations that analyze spatial characteristics and
relationships. If our Elementary teachers are going to introduce our
young mathematicians to the world of space and number through physical
forms, then we must not drop the bucket and let the kids believe that the
learning they gained through their hands on experience is not
sophisticated enough, or the right stuff for really smart kids. I guess I
better qualify all of this harangue with the fact that I am a *Visual
Learner*
and was always the student asking, show me a picture, and I don't see what
that means! Not given the whole picture, I did not understand the
connection between Algebra and Calculus and Geometry. If we better
understand the set of rules that we are using to define our space, we are
better able to analyze its complexity.
Sorry folks... I told you this pushed some buttons!!! :-)
Kathrine Graham in New Mexico
jgraham@nmsu.edu
As a Math/Science teacher at the H.S. level, this discussion
must surely be equivalent to heated debate I imagine taking place over
the value of formal logic courses , or the benefits of Latin as a
language option in the modern High School.. circa 1935 perhaps ?
Lets get real, I found my 2 years of latin probably the most valuable
course of my high school career, for it support of English through
understanding of cognates, and particularly in science, where upon
jargon came to have meaning: I could make out the meaning of science terms
fairly accurately just from recognizing their latin roots. Dozens of other
real benefits could be desribed, depending on a persons interest;
languages,
history, philosophy, religion-mythology,and so on. If one thinks of
subjects naturally integrative of diverse disciplines Latin studies looks
even brighter as a course offering. NOT !
The value of Geometry as a course of study would seem by analogy tohave
little to do with its inclusion or non-inclusion in a curriculum ! I for
one
am tired of looking at H.S. curriculum as certain doses of Algebra,
Biology,
a litle shot of Physics, some hours of instruction in Geometry, Chemistry,
Calculus, lets round things out with a course in Art, or Music. We do want
well rounded intellects. If we can find a way to introduce the geometry
in and inter-disciplinary context; fine, I'll lend my support. Otherwise
we had better do a much better job of selling our courses than the Latin
teachers.
Smile to all :)
Landon
--
------------------------------------------------------------
lwood@ednet1.osl.or.gov || landonwood@delphi.com
SysOp: Grant Union H.S. PSInet || John Day, OR
------------------------------------------------------------
Reply to: RE>Why teach Geometry in HS pro and con
Why we shouldn't teach geo , the proof course, in h.s.:
1. if we want to teach axiomatic reasoning, we'd be better off doing it
with
algebra or something with a smaller more well defined set of axioms.
(after
all, we did get non-euclidean geometry when folks started poking at some of
euclid's givens)
2. All that time could be better spent exploring more informal verison of
geometry that really do build on and extend our spatial reasoning.
3. Algebra and geometry should go hand and hand, not separated. That
would
make for better prep for calculus stuff too, for those who want that
Why we should teach geometry, the proof course, in h.s.:
There is no better defense of it as a h.s. course than Chip Healy's recent
book. It's working title was "build a book geometry", but it's got another
title now. I think it's published by Dale Seymour. Healy, a teacher in a
working class suburb of LA, has kids write their own geometry book. He
starts
them with 3 axioms, and some tools like Geomters Skecthpad. They determine
the
rules of argumentation, standards of proof and collect all the theorems
that
get proven over the course of a year. He writes not only about what the
kdis
do in class, but the remarkable effects this experience has on their lives
outside of school.
Subject: Re: Why teach Geometry in HS?
Before responding to your email, Patrick, a couple of notes:
- I am responding to only you, John Meseke, and you, Patrick Williams,
because you started the debate & wrote the email, respectively. Is that
right? Or should I be responding to everyone on the list? You asked,
John, that we send our messages to you; are you planning to send this out
to everyone eventually? I don't want to inconvenience you by forcing you
to do my forwarding for me, but I don't want to undermine your position as
moderator by sending everything I write to everybody. How exactly does
this work?
- Also I want you both to know that this response is in the spirit of the
"devil's advocate" responses you asked for, John. Please don't take any
of it personally, Patrick.
In defending geometry in high school, Patrick, you wrote of a few
experiences in which geometry helped you (or would've helped your
neighbor) in real life. It seems to me, though, that those experiences
were not related to the geometry I see taught in high school. For
instance, you talked about your dad's wrangling with the city over the
price of his land, but the math it would take to do that wrangling is
taught well before high school (e.g. finding the area of a parcel of land,
or multiplying the $/acre to find its total value). Any elementary school
grad should have those skills without high school geometry. And the other
examples you gave: a neighbor losing his foundation, building a tree-fort,
etc. are examples of engineering, not geometry. Granted, a good deal of
geometry is used in engineering, but I don't think most geometry classes
focus on the skills needed to determine if a foundation will be undermined
by local excavation. Are you saying that we should learn geometry in case
we choose to become engineers, or that we should forsake geometry for
engineering classes in high school which would be more useful? And
speaking of usefulness, how does construction fit with this (by
construction I mean the staple of high school geometry: compass and
straight-edge construction; perpendicular bisectors, circumscribed circles
and the like)? Will something like that ever be useful outside of the
final exam? Or proofs? Personally, I don't think many people use the
geometry we teach in high school in the outside world.
-Dug Steen
**************************************************************************
John Meseke email- jpm56290@uxa.cso.uiuc.edu
University of Illinois at Urbana/Champaign
"Everybody have fun tonight. Everybody wang chung tonight." -Wang Chung
|
|
|
|
